Volume of Geodesic Balls in Finsler Manifolds of Hyperbolic Type

Let ( ) , M F be a compact Finsler manifold of hyperbolic type, and  F M be its universal Finslerian covering. In this paper we show that the growth function of the volume of geodesic balls of


Introduction and Main Results
A Finsler manifold ( ) , M F is called of hyperbolic type, if there exists on the manifold M a Riemannian metric 0 g of strictly negative curvature such that F and 0 g are uniformly equivalent (cf.Definition 2.3).We say that a function :  is of purely exponential type if there exist constants 1 a > and 0 0 r > such that for some constant 0.

h >
The real number h is called the exponential factor of f.In 1969, Margulis (see [1]) proved, for suitable constant 0 h > that Clearly, this result implies purely exponential growth of volume of geodesic spheres.In 1979, Manning introduced a notion of volume entropy g h of a compact Riemannian manifold ( ) , M g as follows (see [2]): if where the limit on the right hand side exists for all p X ∈ and, in fact, is independent of p. Manning showed that, in the case of non-positive curvature, g h coincides with the topological entropy.In 1997, using the notions of Busemann density and Patterson Sullivan measure, G. Knieper proved the following result (see [3]): is a rank-1 compact Riemannian manifold of non-positive curvature and 0 X its universal Rie- mannian covering, there exist constants 0 1 a ≥ and 0 0 r ≥ such that ( ) Let ( ) , M g be a compact Riemannian manifold of hyperbolic type without conjugate points, and X be its universal Riemannian covering.In 2005, we show that the growth function of the volume of geodesic spheres of X is of purely exponential type with the volume entropy g h as exponential factor (see [4]).The main result of this paper is the following: Theorem 1.1.Let ( ) , M F be a compact Finsler manifold of hyperbolic type and F M  be its universal Fin- slerian covering (cf.Definition 2.3).Let ( ) h F be the volume entropy of F (cf. Definition 2.1).Then, the growth function of the geodesic balls of F M  is of purely exponential type with ( ) h F as exponential factor.Theorem 1.1 implies the following Corollary: Corollary 1.2.Let ( ) , M F be a compact Finsler manifold of hyperbolic type and F M  be its universal Finslerian covering.Then, the critical exponent F α (cf.Definition 4.2) of the group of the Deck transforma- tions of F M  is equal to the volume entropy ( ) h F of ( ) , M F .However, from Theorem 1.1, since all compact orientable surfaces of genus greater than one admits a metric 0 g of strictly negative curvature, we deduce the following properties: Corollary 1.3.Let M be a compact orientable surface of genus greater than one, F a Finsler metric on M and F M  be its universal Finslerian covering.Then, the growth function of the geodesic balls of F M  is of purely exponential type with ( ) h F as exponential factor.The paper is organized as follows: in Section 2, we recall some basic facts about the volume entropy of a compact Finsler manifold.Section 3 is devoted to the ideal boundary and the Gromov boundary of the universal Finslerian covering of a Finsler manifold of hyperbolic type.In Section 4, we introduce a notion of quasi-convex cocompact group and we provide the proof of the Theorem 1.1.

The Volume Entropy of a Finsler Manifold of Hyperbolic Type
In this section, we briefly recall some notions from Finsler geometry; see [5] or [6] and the references therein for more details.Let M be a manifold and denote by : TM is positive definite at every point of , M F be a Finsler manifold M. We say that F is uniformly equivalent to a Riemannian metric g, if there is a constant F c such that 1 .
be the universal covering of M. Using the map p, we pull the Finsler structure F back to M  .The resulting F  defines on \ TM  0 a Finsler structure.We denote by Note that if M is compact manifold and F is invariant under the deck transformation Γ then F and g are uniform equivalence.

Ideal and Gromov Boundaries of Finsler Manifolds of Hyperbolic Type
The following theorem is fundamental for the study of the ideal boundary of Finsler manifolds of hyperbolic type.It was proved by Morse in dimension 2 and by Klingenberg in arbitrary dimensions.The fact that the Morse Lemma also holds in Finsler case was first observed by E. M. Zaustinsky (see [7]).Due to Klingenberg (see [8]), the Morse Lemma holds in any dimension.Theorem 3.1.(Morse Lemma, cf.[9]) Let ( ) , M F be a Finsler manifold of hyperbolic type and g 0 be a me- tric of strictly negative curvature on M such that F and g 0 are uniformly equivalent and M  be the universal covering of M. Then there is a constant ( ) > with the following properties.1) for any two points x and y M ∈  , the g 0 -geodesic-segment ( )  from x to y and any F-minimal segment These properties stay hold for F-backward rays and F-minimal geodesics.

Now let ( )
, M F be a compact Finsler manifold of hyperbolic type and F M  be its universal Finslerian covering.Let g 0 denote an associated metric of strictly negative curvature on M. Note that the universal Riemannian covering 0 M  of ( ) 0 , M g is a Hadamard manifold and let denote by ( ) 0 M ∞  its ideal boundary.Two F-forward rays c and c′ are said to be asymptotic if there exists a constant 0 D ≥ such that ( ) ( ) ( ) , where d H is the Hausdorff distance with respect to the distance d F .This defines an equivalence relation on the set of F-forward rays of follows from Morse Lemma that there exists a g 0 -geodesic ray γ such that ( ) ( ) ( ) , where D is the constant in Morse Lemma.Let [ ] c be the equiva- lence class of a F-forward ray c and let [ ] γ the equivalence class of the g 0 -geodesic γ .The map f defined by Let recall now some basic facts about Gromov hyperbolic spaces.Let ( ) , X d be a metric space with a ref- erence point x 0 .The Gromov product of the points x and y of X with respect to x 0 is the nonnegative real number ( ) x y ⋅ defined by: for all x, y, z and every choice of reference point x 0 .We call X a Gromov hyperbolic space if it is a δ-hyperbolic space for some 0 δ ≥ .The usual hyperbolic space n  is a δ-hyperbolic space, where log 3 δ = .More generally, every Hadamard manifold with sectional curvature (see [10] or [11]).
Lemma 3.2.(see [11] or [12]) Let ( ) , X d be a complete geodesic δ-hyperbolic space, x 0 a reference point in X, x and y two points of X.Then x y ∈  , and (see [11] or [12]) Let ( ) , X d be a δ-hyperbolic geodesic space and 1 2 , : , , , X d be two metric spaces.A map In a metric space X, a quasi-geodesic (resp.quasi-geodesic ray) is a quasi-isometric map ). Lemma 3.6.(see [11]) Let 1 X be a metric space and ( ) , X d be a geodesic Gromov hyperbolic space.If there exists a quasi-isometric map , then 1 X is also a Gromov hyperbolic space.Now let X be a Gromov hyperbolic manifold, 0 x a reference point in X.We say that the sequence ( ) x is another reference point in X, ( ) ( ) ( ) ( ) ( ) Then the definition of the sequence that converges at infinity does not depend on the choice of the reference point.Let us recall the following equivalence relation  on the set of sequences of points in X that converge at infinity: is the coset of sequences that converge at infinity.Let X be a simply connected manifold which is a Gromov hyperbolic space.One defines on the set ( ) G X X ∞  a topology as follows (see [11] page 22): 1) if x X ∈ , a sequence ( ) x ∈ converges to x with respect to the topology of X.

2) if ( )
(see [13]) Let X be a δ-hyperbolic space.Then 1) Each geodesic : ∞ × , there exists a geodesic ray γ such that ( )  is well-defined on X and is called the Busemann function for the geodesic c.Lemma 3.9.(see [13]) Let X be a δ-hyperbolic space, and c a geodesic ray with ( ) where c b is the busemann function for the geodesic c and K is a constant depending only on .δ Lemma 3.10.(see [11]) Let 1 X be a metric space and ( ) , X d be a geodesic Gromov hyperbolic space.If there exists a quasi-isometric map , then 1 X is also a Gromov hyperbolic space.Moreover, if the map ( ) ( ) ( )

G X ∞
The following lemma give an homeomorphism between the ideal boundary and the Gromov hyperbolic boundary of Hadamard manifolds: Lemma 3.11.(see [14]) Let X 0 be a Hadamard manifold with sectional curvature for some constant 0 0.
k > There exists a natural homeomorphism ( ) ( ) Lemma 3.11) and the properties of the ideal boundaries, we obtain the following lemma: Lemma 3.12.Let ( ) , M F be a compact Finsler manifold of hyperbolic type and F M  be its universal Fin- slerian covering.Let g 0 be an associated metric of strictly negative curvature on M and 0 M  be the universal Riemannian covering of ( ) 0 , .

M g
We have . Lemma 3.11).On the other hand, the fact that F is uniformly equivalent to a Riemannian metric g 0 implies that F M  is also a Gromov hyperbolic space and Lemma 3.10).Finally, using the construction of the ideal boundary of

The Growth Rate of the Volume of Balls in Finsler Manifolds of Hyperbolic Type
Definition 4.1.Let X be a Gromov hyperbolic manifold with reference point 0 x and Γ be a discrete and infinite subgroup of the isometry group ( ) Iso X of X .For a given point x X ∈ , the limit set ( ) is the set of the accumulation points of the orbit x Γ in ( ) Definition 4.2.Let ( ) , X d be a metric space and Γ be a discrete and infinite subgroup of the isometry group ( ) is called the critical exponent of Γ and is independent of x and 0 x .The subgroup Γ is called of divergence type if the Poincaré series diverge for s α = .The following lemma introduces a useful modification (due to Patterson) of the Poincaré series if Γ is not of divergence type.Lemma 4.3.(see [15]) Let Γ be a discrete group with critical exponent α .There exists a function  which is continuous, nondecreasing and such that  Let now ( ) , M F be a compact Finsler manifold of hyperbolic type, and F M  be its universal Finslerian covering.Let g 0 denote a metric of strictly negative curvature on M. The universal covering 0 M  of ( ) for some constant 0 0 k > .Let Γ be the group of deck transformations of M  and 0 g α be its critical exponent with respect to the metric 0 g .It follows from Theorem 5.1 in [3] that: ( ) ( ) The fact that M is compact implies the existence of a constant 1 Then, the critical exponent F α of Γ with respect to the metric d F belongs to ( ) ( Lemma 4.4.Let ( ) , M F be a compact Finsler manifold of hyperbolic type, F M  be its universal Finslerian covering and Γ be the group of deck transformations of M  .Then 1)

( ) ( )
, For all y M ∈  , we have:  [3]).Since Γ is cocompact, the identity map 0 : Let now ( ) , X g be a Gromov hyperbolic manifold, and Γ be a non trivial subgroup of ( ) Iso X and the limit set ( ) is the subset of X defined by the collection of the images of the geodesics Γ is compact.The following lemma is due to Coornaert (see [13]).Lemma 4.6.Let ( ) , X g be a Gromov hyperbolic manifold with reference point 0 x , and Γ be a quasi- convex cocompact subgroup of the isometry group ( ) Iso X with critical exponent g α .Then, for all x X ∈ , there is a constant 1 x C ≥ such that: ( ) : for all and .
each point p in manifolds of negative curvature and that the function a is continuous.
with centre p and radius r in the universal Riemannian covering X of ( ) geodesic ray satisfying ( ) .
the Poincaré series associated to Γ .The number Γ .There exists a sequence n γ ∈ Γ such that lim .Γ , by the definition there is a sequence ( )

4 )
Let g 0 denotes a metric of strictly negative curvature on M. The universal Riemannian covering 0

FMF
 .This that Γ is a quasi-convex cocompact subgroup of ( )Iso M  .For an orbit x Γ of Γ in F M we consider the map r K defined by:

1 x
be a fixed point in  and put D diam =  .For all γ ∈ Γ , and x ∈  , we have: